1. finance
2. investing
3. stocks

EMBA 807
Corporate Finance
Dr. Rodney Boehme
CHAPTER 5: HOW TO VALUE BONDS AND STOCKS
(Assigned problems are 1, 4, 5, 7, 8, 9, 13, 16, 17, 18, 21, 22, 23, 25, 26, 29, 31, and 33. Omit
the Appendix to this chapter). This notes package contains two Addendums.
I. BOND VALUATION
Bonds are “Fixed Income” securities, since the cash flows that the bondholder will
receive have been fixed or prespecified in the bond contract.
The current fundamental or intrinsic value of a bond (or any other financial asset)
is equal to the Present Value or PV0 of all future expected cash flows.1
An investor’s actual return on a bond, for any holding period, comes in two forms:
(1) the coupon yield (from the payment of coupon interest)
(2) the capital gain (bond’s change in price)
Zero Coupon Bonds: a zero coupon bond will pay its stated face or par value
at maturity. It pays no other future cash flows during its life. Zeroes are also
known as pure discount bonds. The return here comes entirely as a capital gain.
Example: A zero coupon bond will mature exactly 2 years from today. It was
sold or issued by the U.S. Treasury and therefore is free of default risk. The par or
face value is $100. This bond currently sells for $84.17 in the market. What
annual rate of return does the market expect on this bond?
From Chapter 4, we know: PV0 = FVn/[1 + r]n. Here, PV0=$84.17, FV2=$100, and
n=2 years. We need to solve for r. A time line of this bond is shown below:
t=0
t=1
PV0 = 84.17
t=2
FV2 = 100
PV0 = FVn/[1 + r]n → r = [FVn/PV0]1/n – 1 = [100/84.17]0.5 – 1 = 0.09 or 9.0%
Actually, anytime we calculate the annual yield r by using the current bond price
and future payments, the r is referred to as the Yield-to-Maturity or YTM.
What will happen to this bond’s price tomorrow if the current market rate of
interest or YTM on this and similar two-year bonds falls from 9.0% to 8.8%?
1
Intrinsic value refers to a private estimate of value – a private estimate of Present Value. This is not the same
concept as market value, which refers to the current price at which a bond or stock is trading for.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
PV0 = FVn/[1 + r]n = 100/[1+0.088]2 = $84.48. Here, the bond is now priced to
yield the new lower market rate of interest of 8.8% per year. As yields fall, bond
prices rise. Bond yields (interest rates) and prices are always inversely related.
Coupon Bonds: most bonds pay coupon interest over their lives. Most coupon
interest payments are paid every six months or semiannually. However we will
cover annual coupon paying bonds first. We begin with an example.
Example: A bond was issued 10 years ago as a 20-year bond. Today, it has 10
years remaining until maturity. The face or par value of the bond is $1000. The
bond will pay an annual coupon interest payment equal to 9% of the par value.
Each year this bond will pay (0.09)(1000) = $90 as a coupon interest payment.
Over its remaining 10-year life, the bond will thus pay ten $90 coupons and one
amount of the $1000 par at maturity.
Note: the coupon rate of 9% only determines the amount of the annual coupon
payment. The coupon rate is completely independent of the actual YTM or current
market rate of return on this bond. This bond will pay the $90 coupons, regardless
of what happens to the YTM. After all, this is a fixed income security.
At this time, market conditions are such that investors expect to earn a YTM of
8.5% on this and similar bonds. A time line of the bond is shown.
t=0
t=1
t=9
t=10
YTM = 8.5%
$90
$90
$90 +
$1000 par
The ten $90 coupons represent an ordinary annuity (as covered in Chapter 4) of ten
$90 cash flows. Then there is one lump sum of $1000 at maturity. We calculate
the Present Value or PV0 of all these payments (the annuity and lump sum).
1

1
PAR
PV0 = C  +
n
 r r (1 + r )  (1 + r )n
 1

1
1000
PV0 = 90 
 +
10
 0.085 0.085(1 + 0.085)  (1 + 0.085)10
PV0 = (90)(6.561348) + 1000/(1+0.085)10
PV0 = 590.52 + 442.29 = $1032.81
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
The coupon and par payments are worth $590.52 and $442.29 today (t=0),
respectively. Thus investors are willing to pay $1032.81 for this bond investment
today and expect to receive a series of eleven future payments; ten annual coupon
payments of $90 each and one $1000 amount ten years from today.
To work this problem on a financial calculator, these are the steps:
FV = $1000 (the par value)
I = 8.5% (the current YTM)
N = 10 (number of coupons and number of periods until the par is received)
PMT = $90 (the amount of the annual coupon)
P/Y = 1 (since periods and payments are made annually in this case)
The answer is PV –$1032.81, and thus the price is $1032.81. When the
market price is greater than the par value, we say that the bond is selling at a
premium (to par value).2
Bond Prices and Interest Rate Changes:
The previous bond is worth $1032.81 today if the market requires a YTM=8.5%.
Market rates of interest change continuously as economic conditions change.
When interest rates or yields fall (rise), bond prices rise (fall). In all cases, bond
yields and prices are inversely related. The only way to obtain a lower (higher)
yield or return on any fixed income investment is to pay more (less) for it.
In the last example, assume that the current market 8.5% YTM: (1) falls to 8%; (2)
rises to 9%; and (3) rises to 9.5%.
(1) YTM falls from 8.5% to 8.0%
FV = $1000
I = 8.0% (the new YTM)
N = 10
PMT = $90
P/Y = 1
The answer is PV –$1067.10, and thus the new price is $1067.10. The price
rises as the market YTM falls.
(2) YTM rises from 8.5% to 9.0%
2
Note that in exactly one year, the bond has nine remaining years. If the market YTM is still 8.5%, then the price
next year will be $1030.60. Thus during course of one year, the bond pays $90 in coupons and also falls in price by
$2.21. An investor’s holding period return for the year is thus [1030.60-1032.81+90]/1032.81 = 87.79/1032.81 =
0.085 or 8.5%, an amount that is equal to the YTM.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
FV = $1000
I = 9.0% (the new YTM)
N = 10
PMT = $90
P/Y = 1
The answer is PV –$1000, and thus the new price is $1000. The price falls
as the market YTM rises. When the YTM and coupon rate become equal,
the bond price is equal to the par or face value.
(3) YTM rises from 8.5% to 9.5%
FV = $1000
I = 9.5% (the new YTM)
N = 10
PMT = $90
P/Y = 1
The answer is PV –$968.61, and thus the new price is $968.61. When the
market price is less than the par value, we say that the bond is selling at a
discount (to par value).
Semiannual Coupon Bonds: most bonds pay coupon interest every six
months. However, coupon rates are always stated on an annual basis. If a $1000
par bond states a 10% annual coupon rate, to be paid semiannually, that means that
10%/2 = 5% of the par value or $50 is paid every six months as a coupon payment.
Example: A U.S. government treasury bond will mature exactly 11.5 years from
today. This bond was likely first issued as a 30 year bond 18.5 years ago. The par
value is $1000. The annual coupon rate is 14% and is paid out semiannually.
Today, investors expect this and similar bonds to yield a YTM of 8% per year.
Note: here, the YTM=8%, means 8% annual nominal, compounded semiannually.
The actual effective interest rate here is something greater than the YTM of 8%.
However, the Wall Street Journal and other financial sources will quote the YTM.
This bond pays (14%/2)(1000) = $70 every six months as a coupon payment
The remaining number of periods in this bond’s life is (11.5)(2) = 23 semiannual
periods. This bond’s time line is shown below.
t=0
t=0.5 year
$70
t=1 year
$70
t=11 years
$70
Page 4
t=11.5 years
YTM=8% per year
$70 +
$1000 par
EMBA 807
Corporate Finance
Dr. Rodney Boehme
This bond’s current value is found as follows (n=11.5 years and m=2):


1
1
PAR

+
PV0 = C
n⋅m 
n ⋅m
YTM

 1 + YTM m
m YTM m 1 + YTM m
(
)
(
)
 1

1
1000
PV0 = 70
 +
23
0.04
0.04(1 + 0.04 )  (1 + 0.04 )23

PV0 = (70)(14.856842) + 1000/(1+0.04)23
PV0 = 1039.98 + 405.73 = $1445.71
Calculator method (one of two ways)3:
FV = $1000 (the par value)
I = 8% (the annual YTM)
N = 23 (number of 6-month or semiannual coupon periods until the par is
received and the number of actual $70 coupons)
PMT = $70 (the amount of the semiannual coupon)
P/Y = 2 (since periods and payments are semiannual in this case; the
calculator divides the YTM by m=2 and uses 4% as the effective 6month rate to work this problem)
The answer is PV = –$1445.71, and thus the price is $1445.71
Note: working this problem as having annual coupons of $140 each will always
produce the wrong answer!
3
The alternate calculator method is to let P/Y=1 and I=4%. Everything else stays the same.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
II. STOCK VALUATION
Current stock prices reflect today’s expectations of the future cash flow
performance of a corporation, as well as its expected risk. Expectations
concerning the future can never be proven in the present.
We will assume that the firm accepts all new projects that increase the value of the
firm, i.e., projects having a positive NPV. The excess cash that remains that can be
paid out to the shareholders is referred to as Free Cash Flow to Equity or FCFE.
Here, we will assume that the FCFE is all paid out as a cash dividend. In reality,
today firms pay out the FCFE by (1) dividends and (2) stock repurchases.
An analyst must forecast the firm’s ability to pay out cash to stockholders in the
future. The fundamental or intrinsic value of a stock is defined as the Present
Value of all future expected FCFE (again, assume it is paid as dividends) that will
be paid out. Observe the following time line of expected dividends.
t=0
t=1
t=2
t=3
t=4
D0
D1
D2
D3
D4
We usually assume that the D0 (if it exists) has just been paid out and is no longer
part of the firm. The fundamental stock price P0 can be modeled as follows4:
P0 =
∞
D
∑ (1 + tr )t
, which is also expressed as
P0 =
t =1
D1
(1 + r )
1
+
D2
(1 + r )
2
+
D3
(1 + r )
3
+ ... .. +
Dt
(1 + r )t
+ ......
The term r represents the market’s required rate of return on the stock. In Chapter
10, we will learn how the r is determined.
Mature or Constant Growth Stocks:
This concept was first introduced in Chapter 4. With regard to the common stock
of a mature firm, the dividend stream is expected to grow at a constant rate g as
time passes. Everything associated with the firm is also expected to grow at the
same rate g, including earnings, sales, cash flows, and the stock price.
Assume that a mature firm has just paid out dividend D0=$5 per share to its
common stock. This dividend stream is expected to grow at g=5% per year.
Dividends D1 and D2 paid exactly one and two years from today are expected to be
the following:
4
A better term would be V0 for value, rather than P0 which is interpreted as price.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
D1 = D0(1+g) = 5(1+0.05) = $5.25
D2 = D0(1+g)2 = 5(1+0.05)2 = $5.5125
The constant growth model is: Pt = Dt+1/(r-g). The stock’s value today, just after
the D0 has been paid is estimated to be: P0 = D1/(r-g).
The required annual rate of return on this stock is r=14%. The constant growth
model introduced in Chapter 4 can be used to estimate the value of this stock.
P0 = D1/(r-g) = 5.25/(0.14-0.05) = 5.25/0.09 = $58.33
Thus the Present Value of all expected future cash flows is estimated to be worth
$58.33 today. Does this mean that the stock actually sells for $58.33 in the market
today? If the analyst is correct about the fundamental value being $58.33, then if
the stock currently sells for $50 it is undervalued in the market.
Also, according to this model, the stock’s value next year, just after the D1=$5.25
is paid out should be:
P1 = D2/(r-g) = 5.5125/(0.14-0.05) = $61.25. However, note that the following
approach also works:
P1 = P0(1+g) = 58.33(1+0.05) = $61.25
Changing the r or g in the constant growth model:
What happens to the current value P0 above if the required rate of return decreases
to r=12%?
P0 = D1/(r-g) = 5.25/(0.12-0.05) = $75.00. Thus the stock price should increase
from $58.33 to $75.00.
If the required return r increases (decreases), the stock price decreases (increases).
If the cash flow growth rate g increases (decreases), the stock price increases
(decreases).
Extensions of the constant growth model:
Here, we rearrange the model to explore other aspects.
P0 = D1/(r-g) → rearrange to obtain → r = D1/P0 + g
Using data from the prior example; P0=$58.33, D1=$5.25, g=5%, and r=14%.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
r = D1/P0 + g = (5.25/58.33) + 0.05 = 0.09 + 0.05 = 0.14 or 14.0%
This analysis tells us that the r=14% required rate of return on the stock comes as:
(1) a 9% dividend yield
(2) a 5% capital gains yield, e.g., an increase in share price of g=5% from
P0=$58.33 to P1=$61.25.
Stocks with multiple or nonconstant growth stages:
The constant growth model; P0=D1/(r-g), applies where the cash flows following D1
at t=1 are expected to grow at a constant annual rate g.
What about a firm that, like most firms, for the next 5 to 15 years, is not expected
to grow at a constant rate? This same analogy can certainly be extended to firms
that currently pay no dividends at all.
If the firm pays no dividends today (any cash flow is going toward reinvestment in
the firm), then we must assume the firm will begin paying dividends at some time
in the future and eventually mature, and thus grow at roughly the same rate as the
overall economy.
Also, if the firm currently pays dividends, but these dividends are not expected to
grow at constant rates during the near future, we still must assume that the firm
does mature at some future point. A model such as the following is used, where at
some time in the future, we will assume maturity and constant growth. Here, all
dividends following the dividend at time t+1 grow at the annual rate g.
P0 =
D1
+
D2
+
D3
(1 + r )1 (1 + r )2 (1 + r )3
+ ... .. +
Dt
(1 + r )t
D  1 
+  t +1  
t
 r - g   (1 + r ) 
Example: Today, Cirrus Corp. is a growing firm that pays no dividends. It has
earnings today, but it reinvests all of its earnings into new growth projects. You
expect that Cirrus will pay dividends in the future, beginning with year 4 (t=4).
The expected or forecasted dividends are D0=D1=D2=D3=0, D4=$0.50, D5=$0.65,
D6=$0.80, D7=$0.90, and D8=$1.00. All dividends after year t=8 are expected to
grow at a constant annual rate of g=6%. The required rate of return on the stock is
r=10% per year. The time line is shown below:
t=0
t=1
t=2
t=3
t=4
t=5
t=6
t=7
t=8
t=9
D0=0
D1=0
D2=0
D3=0
D4=0.50
D5=0.65
D6=0.80
D7=0.90
D8=1.00
D9=1.06
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
We next estimate today’s intrinsic value of this stock. The valuation model can be
set up as shown below. Note that the term, D8/(r-g), is very critical. This is the
estimated stock price exactly 7 years from now. While D9 is shown on the time
line, it is not needed.
P0 =
P0 =
D4
+
D5
+
D6
(1 + r )4 (1 + r )5 (1 + r )6
0.50
0.65
+
+
+
 D  1 
+  8 

(1 + r )7  r - g   (1 + r )7 
D7
0.80
(1 + 0.10 )4 (1 + 0.10)5 (1 + 0.10)6
+

1
 1.00  
+


(1 + 0.10 )7  0.10 - 0.06   (1 + 0.10)7 
0.90
P0 = 0.3415 + 0.4036 + 0.4516 + 0.4618 + [25][0.5132] = $14.49
This private estimate of the intrinsic value of Cirrus stock is P0 = $14.49 per share.
If the actual market price P0 is below or above $14.49 then you may want to buy or
short the stock, respectively.5 The Cirrus stock price at t=7 years is expected to be
P7 = D8/(r-g) = $25.00, just after the dividend of D7=$0.90 has been paid out. The
P7 = $25 is deemed the terminal price (value when g becomes constant).
Example: XYZ Corp., a growth firm, is expected to pay its first dividend exactly
18 years from today. XYZ is forecasted to pay out nothing before that time. An
analyst has estimated the following: D18=$6.00, r=14%, and g=7%. Calculate
today’s estimated intrinsic value.
t=0
t=1 year
t=2 years
$0
$0
t=17 years
$0
t=18 years
$6.00
Step 1: find the estimated stock price exactly 17 years from today.
P17 = D18/(r-g) = 6/(0.14-0.07) = $85.7143
Step 2: Find today’s (t=0) intrinsic value. Discount the P17 price back to today to
calculate P0.
P0 = P17/(1+r)17 = 85.7143/(1+0.14)17 = $9.24
5
Short selling is selling a stock that you do not own, hoping that it will fall in price. Essentially, these shares are
borrowed and identical shares must later be bought and returned to the owner. The goal is to buy back the shares at
lower price than they were sold. In going short, you hope to sell high, and buy low. This is opposite the position of
being long or owning the stock. In going long on a stock (having purchased a stock) you hope to buy low and sell
high.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
Estimating the permanent or long-run growth rate “g”:
We now discuss two common methods of estimating the permanent annual growth
rate g.
1. g = expected annual inflation + expected annual real growth in GDP
For example, let expected inflation and GDP growth be 3% and 2.5%,
respectively. The growth rate is then expected to be g = 3% + 2.5% = 5.5%.
In the long run, it is unrealistic to assume that any firm can sustain a growth
rate that exceeds the Gross Domestic Product growth of the economy. Short
run growth rate estimates may certainly exceed the GDP growth rate.
2. g = [ROE x b]
ROE is defined as the average future expected annual Return on Equity on the
firm’s capital expenditures. If a $100 real investment generates $16 per year
forever for shareholders, then the investment has an annual ROE = 16/100 =
16%. The firm’s retention or plowback ratio b is defined as the proportion of
economic earnings “E” that the firm retains and reinvests (invest here means net
capital expenditures, or total capital expenditures minus depreciation). The
firm thus has a payout ratio of (1-b) to the stockholders.
• If ROE = 16% and b = 0.40, then g = (0.16)(0.4) = 0.064 or 6.4% per year.
• Using the constant growth model, you can replace “g” with (ROE)(b) and D1
with (1-b)(E1) to obtain the following variant of the constant growth model:
P0 =
(1 - b)E1
r - (ROE)(b)
Example: Cash Cow Inc., a mature firm, has r=10%, next years earnings are
expected to be E1=$2 per share, retention ratio b=0.4, and expected ROE=16% on
its corporate investments. What is the intrinsic value of Cash Cow today?
P0 =
(1 - 0.4)(2.00)
1.20
=
= $33.33
0.10 - (0.16)(0.4) 0.10 - 0.064
Corporation Value and Growth Opportunities:
Any firm’s current value can be decomposed into two portions:
Total firm value = PV of CFs from current operations
+ PV of future NPV growth activities
The first item above is often called the value of assets in place.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
Example: An existing firm has no growth opportunities. The number of existing
shares is 20 million. The firm invests just enough to keep the current assets in
production (no net investment occurs, as total investment then equals the
depreciation). In this case, the earnings equal the dividends paid and will be $100
million per year forever (a perpetuity), so g=0%. Required rate of return on the
stock is r=15% per year.
First, estimate the value of the existing shares of stock:
P0 = D1/(r-g) = (100M/20M)/(0.15-0) = 5/0.15 = $33.3333 per share
Now a new project unexpectedly comes along. It costs $15 million today and
generates cash flows of $5 million per year forever, beginning at t=1.
NPVproject = -15 + 5/0.15 = -15 + 33.3333 = $18.3333 million.
Project’s impact on existing stock price value: 18.3333M/20M = $0.9166 per
share
New stock price: P0 = 33.3333 + 0.9166 = $34.25 per share
Price To Earnings or P/E Ratios:
We revisit the constant growth model. It is rearranged to solve for the P/E ratio.
P0 =
(1 - b)E1
r - (ROE)(b)
→
P0
(1 - b)
=
E1 r - (ROE)(b)
Note: P0/E1 = (1-b)/(r-g). What about a very large mature firm that sells for a P/E
of 60 to 80. Let b=0.4 and r=10%. If the P/E is 70, then what is the market’s
consensus estimate of g?
P0/E1 = (1-b)/(r-g) → 70 = (1-0.4)/(0.10-g) → 0.10 – g = 0.6/70 → g = 9.14%
This growth estimate is certainly unrealistic for a large firm, especially if the
nominal GDP growth of the economy is 5 to 6% per year. What if the market
begins to price the stock using a more realistic growth estimate of g=6%?
P0/E1 = (1-b)/(r-g) → P0/E1 = (1 – 0.4)/(0.10 – 0.06) = 15, which is close to the
historical P/E average for mature stocks and the overall stock market (such as the
S&P 500 index).
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
ADDENDUM 1 TO CHAPTER 5
This section presents a more integrated treatment of Corporate Finance and stock
valuation.
First, we must define the following items:
ROE
The firm’s expected average Return On Equity on its future real
investments or capital expenditures.
r
The market’s required rate of annual return on the common stock,
based upon its level of risk; to be formally illustrated in Chapter 10.
b
Proportion of earnings (net income) that is reinvested into the firm as
Net Capital Expenditures.6 b is also known as the retention ratio.
g
Permanent growth rate of a mature firm. Defined here as
g = (ROE)(b).
CE
Capital Expenditures (the new positive NPV investments)
INT
Interest expense
T
Corporate income tax rate
DEP
Depreciation (never a cash flow, it is a non-cash expense)
FCFE
Free Cash Flow to Equity, the excess cash flow that can be paid out to
the stockholders.
EBITDA Earnings before Interest, Taxes, Depreciation, and Amortization;
basically revenue minus operating costs.
Assume that we have a mature firm that is 100% financed by equity or common
stock, and thus we allow the interest expense in this example is zero. The current
figures apply for the fiscal year that has just ended.
Let CE = $6000, DEP = $5000, and EBITDA = $9167. Also, let r = 10%,
ROE = 15%, b = 40%, T = 40%, and 10,000 shares of common stock exist.7
We want to estimate the fundamental or intrinsic value of this firm. We will just
assume that the current FCFE0 has just been paid out as a share repurchase and/or
cash dividend. We further simplify this example by assuming that there is no
investment in Net Working Capital (short term assets as shown in Chapter 7)
6
If total capital expenditures are equal to depreciation, then there are no net capital expenditures. Net capital
expenditures are equal to total capital expenditures minus depreciation.
7
The r = 10% represents the firm’s cost of equity capital.
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Corporate Finance
Dr. Rodney Boehme
FCFE0 ≡ [EBITDA – DEP – INT][1 – T] + DEP – CE
FCFE0 = [9167 – 5000 – 0][1 – 0.40] + 5000 – 6000
FCFE0 = 2500 + 5000 – 6000 = $1500 (the $2500 amount is the Net Income)
Net Capital Expenditures ≡ Total CE – DEP = 6000 – 5000 = $1000
Retention ratio = b = [Net CE/Net Income] = 1000/2500 = 0.40 or 40%, note that b
was already given but this is where it originates.
Calculate this mature firm’s permanent or long-run annual growth rate g:
g = (ROE)(b) = (0.15)(0.4) = 0.06 or 6% per year. Note that new Capital
Expenditures are expected to earn an average return of 15% per year, while r =
10%, meaning that the capital expenditures are expected to generate a higher
return than the cost of capital.
In fact, all of the above figures used in the calculation of FCFE0 are forecasted to
grow by g = 6% per year.
Next year’s (t = 1) forecasted FCFE1, based on what we assume today:
FCFE1 = FCFE0(1 + g) = 1500(1 + 0.06) = $1590
Now estimate the current (t=0) value of this firm’s equity:
Value = P0 = FCFE1/(r – g) = 1590/(0.10 – 0.06) = $39,750
Fundamental value is thus [$39,750/10,000 shares] = $3.975 per share
If the current market price were actually less than (greater than) $3.975, then we
would believe that this stock is undervalued (overvalued).
Be aware that this is only a forecast or estimate of the current value. We don’t
know any of these numbers or parameters in the above models with certainty.
RWJ textbook: the text gives P0 = D1/(r – g). This is technically accurate if the
firm pays out all of its FCFE as a dividend; however, this scenario is not likely.
Calculate the P/E ratio, defined here as P/E = P0/NI1:
P0/E1 = 39,750/[(2500)(1.06)] = 15 for the leading P/E ratio. This firm is
expected to trade at a price that is 15 times next year’s earnings or Net Income.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
Growth versus No Growth:
Assume that the firm has no growth opportunities. In this scenario, CE = DEP, for
each year as the firm invests just enough to offset the depreciation, i.e., there are no
Net Capital Expenditures and also g = 0 and b = 0.
FCFE0 ≡ [EBITDA – DEP – INT][1 – T] + DEP – CE
FCFE = [9167 – 5000 – 0][1 – 0.40] + 5000 – 5000 = $2500 for every year
Value as no growth = FCFE/(r – g) = 2500/(0.10 – 0) = $25,000
The previous value of the firm, assuming 6% annual growth, was $39,750. Now
we can compare the growth versus no growth values of this firm. Any firm’s value
consists of the following:
Total Value = PV(no growth) + PV(future NPV growth opportunities)
39,750 = 25,000 + NPVGO
Thus the NPVGO = $14,750
Note that if r = ROE, then future investments have zero NPV and will add no value
over the no growth value. Growth opportunities only add value if they have a
positive NPV, here meaning that ROE > r. Conversely, if ROE < r, then new
investments will destroy value!
Also note one very important item from these Chapter 5 notes: equity valuation
takes into consideration all of today’s expectations of future performance.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
ADDENDUM 2 TO CHAPTER 5
Corporate valuation using Free Cash Flows (FCFs):
The most commonly used valuation tool by financial analysts the Free Cash Flow
(FCF) valuation model, which is somewhat different from the FCFE model
covered in Addendum 1, although there are many similarities. The FCFE model is
used to value only the firm’s equity, while the FCF model is used to value the
entire corporation (the sum of debt and equity). Once the entire corporate value is
estimated, then the debt value is subtracted out, and then what remains is thus the
estimate of the equity value.
At this point in the course, there are some disadvantages with covering the Free
Cash Flow or FCF model. First, the discount rate or cost of capital that must be
used is the weighted average cost of capital or WACC — which is not fully
covered until Chapter 12.8 Free Cash Flow (FCF) is roughly defined the cash that
the firm can pay out to its investors, both stock and debt owners, after the firm has
met its obligations and investment needs.
The FCF can be expressed by the equation shown below — it is important to
compare this FCF tool below to the Capital Budgeting cash flow analysis that we
will cover in Chapter 7, as the FCF method values the entire firm, while Capital
Budgeting methods in Chapter 7 value a single project.
FCFi = [EBIT][1 – tax rate] + Depreciation – Capital Expenditures
– ∆Net Working Capital – Principal Repayments + New Debt Issues
The most common practice is to estimate ten individual annual FCFs and then
assume maturity or constant annual growth following year 10 at the rate g. Thus
the FCF valuation model most commonly used appears below.
V0 =
FCF1
(1 + wacc)
1
+
FCF2
(1 + wacc)
2
+
FCF3
(1 + wacc)
3
+ ... .. +
FCF10
(1 + wacc)
10

 FCF11  
1
+



10
 wacc - g   (1 + wacc) 
The constant growth model as used above obtains V10=FCF11/(wacc-g). V0 is the
total intrinsic value (enterprise value or equity plus debt) of the firm. Subtract the
current debt value from V0 to obtain the estimated equity value.
8
WACC represents a weighted average cost of capital for the firm, or what the firm’s current or existing mix of both
debt and equity financing currently cost the firm. In Chapter 12, we will cover this concept in detail.
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EMBA 807
Corporate Finance
Dr. Rodney Boehme
For firms that need infusions of cash for the next few years, before the FCFs
(hopefully) eventually become positive, the FCF model allows you to use negative
FCFs.
Example: Today, Trillium Corp. is a growing firm that pays no FCF. It has
earnings today; however, after reinvesting all of its earnings into new growth
projects, it must still borrow additional cash to meet its investment needs. You
expect that Trillium will eventually pay positive FCFs in the future, beginning with
year 4 (t=4). The expected or forecasted FCFs (all given in millions of US$) are
FCF1=-$0.80, FCF2=-$0.50, FCF3=-$0.20, FCF4=$0.50, FCF5=$0.65, FCF6=$0.80,
FCF7=$0.90, FCF8=$1.00, FCF9=$1.10, and FCF10=$1.20. All FCFs after year
t=10 are expected to grow at a constant annual rate of g=6%. The WACC of this
firm is 10% per year. The firm has $2 million of debt outstanding (at current
market value).
We next estimate today’s intrinsic value of this firm. The valuation model can be
set up as shown below. Note that the terminal value term, FCF11/(r-g), is very
critical, as this is the estimated firm value exactly 10 years from now.
V0 =
FCF1
+
FCF2
+
FCF3
(1 + wacc)1 (1 + wacc)2 (1 + wacc)3
V0 =
+
+ ... .. +
FCF10
(1 + wacc)10

 FCF11  
1
+


10 
 wacc - g   (1 + wacc) 
- 0.80
- 0.50
- 0.20
0.50
0.65
0.80
+
+
+
+
+
2
3
4
5
(1 + 0.10) (1 + 0.10) (1 + 0.10) (1 + 0.10) (1 + 0.10) (1 + 0.10)6

0.90
1.00
1.10
1.20
1
 1.272  
+
+
+
+

7
8
9
10
10
(1 + 0.10) (1 + 0.10) (1 + 0.10) (1 + 0.10)  0.10 - 0.06   (1 + 0.10) 
V0 = – 0.7273 – 0.4132 – 0.1503 + 0.3415 + 0.4036 + 0.4516 + 0.4618 + 0.4665
+ 0.4665 + 0.4627 + [31.80][0.3855] = $14.0237 million
This private estimate of the intrinsic value of Trillium is V0 = $14.0237 million.
Since this firm has $2 million of debt outstanding (at current market value), the
firm’s equity is thus estimated to have an intrinsic value of $14.0237 – $2 =
$12.0237 million. If there are 100,000 shares of common stock, then the stock is
estimated to be worth $120.24 per share.
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